Let us put ourselves in Yuki’s shoes for a minute, we know that we will do an error in the future but is unable to stop the error from actually occurring. As time passes by, she get closer to committing the error and her foreknowledge of the future is gone. Soon, Yuki knows something will cause her to change her current mindset and make her go astray. That is probably a very scary possibility and she must have been trying very hard to prevent it to no avail. And then, time began to continually loop around her, no one is actually getting hurt and the IDTE is still getting their data. Thus, Yuki allows the loop to continue with minimum interference in her part. No matter how much boredom Yuki will get from undergoing this loop, it is familiar and safe and as long that time is looping disappearance won’t happen.

The sad thing about this is it’s because of endless eight that Yuki got those errors that caused said disappearance. Thus, by trying to prevent disappearance from happening endless eight occurred which made disappearance happen. When you think about it, is it no wonder Yuki sealed the ability to synchronize...

That's a very interesting way of looking at it. I'd like to believe it because it's a very cool way of interpreting things, but I also think it's flawed. Remember, E8 isn't some black hole for Yuki where she doesn't know what will happen or how long it will last. Her memories persist throughout each iteration, so it only makes sense that she could synchronize with herself over those 600 years.

Yuki's not trying to change or prevent the future. She never does that. In fact, like the time travelers, she feels forced to act a certain way because she knows that's how she'll act. She knows she'll go through 15000-something iterations of summer, so when it gets to that point, she just accepts it.

We think it's bad to go back in time and screw up the past, because it might change the present and the future. We generally have no similar reservations about changing the future, because hey, it's the future. Who know what'll happen? But in the Haruhiverse, where we KNOW future time travelers exist, everyone is in the "past" for those people. Yuki can't allow herself to change their present, because it could have serious repercussions on her present.

It's weird, because there's that concept of predestination that defines character growth. Nothing Yuki does is anything but living out how she "knows" things will happen. She's only developing because it's predetermined that she'll develop, and she only lets E8 continue because she knows she'll let E8 continue. Even if she has feelings about it and about her future, they don't come into play.

Concerning Yuki being Rei 2.0 (a discussion from the first page):
Yuki is undoubtedly based on Rei (look no further than the name), but she's not really "Version 2.0." As someone else said, she's just a different take on the "emotionless girl" trope. Rei is the way she is because she's kept secluded, with limited human contact. She acts like a robot because she's treated like one, and she changes (quite quickly, actually) once she's thrust into real society. I have an image on my computer with all sorts of instances of Rei showing emotion (ironically titled "Rei Ayanami is an Emotionless Doll"). She's completely human and she reacts to many things just as an other human would.

Yuki, on the other hand, is not human. She's a computer, and honestly, the fact that she develops feelings at all (regardless of what Kyon thinks of the matter in Disappearance) is completely ridiculous. Compare Asakuru. She's immersed in the world just as much as Yuki is, and she has no trouble dealing with people. But in truth, her emotions are just what one might expect from a computer: pure facades, and nothing more. She uses emotions as a tool, nothing more. A computer picking up emotions because it's surrounded by humans is as ridiculous as a computer generating fiction because there are a bunch of e-books on its hard drive.

Rei makes sense as a character. Yuki does not. But there's a very crucial reason why Yuki does not, and it's because she's a better character that way. She's better for the stories, and she's better for the audience. I like her the most of all the characters because of it.